Question 4156: Positive-Mass Value Recovered from a Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 7, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-19/9. Find that slot’s integer value, written as p/1.

• G001 — S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 7, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-19/9. Find that slot’s integer value, written as p/1.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-expectation

Step-by-step derivation

total mass numerator := sum(6, 2, 1)

total mass numerator = 9

indexed masses=[6, 2, 1];D=9

probabilities=[2/3, 2/9, 1/9]

known numerator0 := mul(6, -5)

known numerator0 = -30

known numerator1 := mul(2, 7)

known numerator1 = 14

known weighted numerator := sum(-30, 14)

known weighted numerator = -16

target expectation numerator := mul(-19/9, 9)

target expectation numerator = -19

unknown weighted contribution := sub(-19, -16)

unknown weighted contribution = -3

recovered unknown value := div(-3, 1)

recovered unknown value = -3

recovered weighted numerator := mul(1, -3)

recovered weighted numerator = -3

complete reconstructed numerator := sum(-16, -3)

complete reconstructed numerator = -19

reconstructed expectation := div(-19, 9)

reconstructed expectation = -19/9

k=2;mass_k=1>0;target=-19/9

unique recovered value=-3;substitution mean=-19/9

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute reconstructed expectation with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute complete reconstructed numerator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute recovered weighted numerator with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute recovered unknown value with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute unknown weighted contribution with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute target expectation numerator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute known weighted numerator with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute known numerator1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute known numerator0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute total mass numerator with exact rational arithmetic, not rounded decimals.

Result

-3/1

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.